What Is The Approximate Value Of { X $}$ In The Equation Below?${ \log _{\frac{3}{4}} 25 = 3x - 1 }$A. { -3.396$}$B. { -0.708$}$C. ${ 0.304\$} D. ${ 0.955\$}

What is the Approximate Value of $x$ in the Equation?

In this article, we will explore the concept of logarithms and how to solve equations involving logarithms. We will use the given equation $\log _{\frac{3}{4}} 25 = 3x - 1$ to find the approximate value of $x$. This equation involves a logarithm with a base of $\frac{3}{4}$ and a result of $25$. We will use the properties of logarithms to solve for $x$.

A logarithm is the inverse operation of exponentiation. In other words, if $a^b = c$, then $\log_a c = b$. The logarithm of a number is the exponent to which the base must be raised to produce that number. For example, $\log_{10} 100 = 2$ because $10^2 = 100$.

To solve the equation $\log _{\frac{3}{4}} 25 = 3x - 1$, we can start by isolating the logarithm term. We can do this by adding $1$ to both sides of the equation:

$$\log _{\frac{3}{4}} 25 + 1 = 3x$$

Next, we can use the property of logarithms that states $\log_a b + \log_a c = \log_a (bc)$. We can rewrite the equation as:

$$\log _{\frac{3}{4}} (25 \cdot 10) = 3x$$

Since $25 \cdot 10 = 250$, we can simplify the equation to:

$$\log _{\frac{3}{4}} 250 = 3x$$

Using a Calculator to Find the Logarithm

To find the value of $\log _{\frac{3}{4}} 250$, we can use a calculator. We can enter the equation into the calculator and press the "log" button. The calculator will give us the value of the logarithm.

Using a calculator, we find that $\log _{\frac{3}{4}} 250 \approx 2.708$.

Now that we have the value of the logarithm, we can solve for $x$. We can do this by dividing both sides of the equation by $3$:

$$\frac{\log _{\frac{3}{4}} 250}{3} = x$$

Substituting the value of the logarithm, we get:

$$\frac{2.708}{3} = x$$

Simplifying, we get:

$$0.896 = x$$

The value of $x$ is approximately $0.896$. However, we are given four possible answers: $-3.396$, $-0.708$, $0.304$, and $0.955$. We need to determine which of these answers is closest to the value of $x$.

To compare the answers, we can calculate the absolute difference between each answer and the value of $x$. The answer with the smallest absolute difference is the closest to the value of $x$.

The absolute differences are:

• $|-3.396 - 0.896| = 2.5$
• $|-0.708 - 0.896| = 0.188$
• $|0.304 - 0.896| = 0.592$
• $|0.955 - 0.896| = 0.059$

The smallest absolute difference is $0.059$, which corresponds to the answer $0.955$.

In this article, we used the properties of logarithms to solve the equation $\log _{\frac{3}{4}} 25 = 3x - 1$. We found that the value of $x$ is approximately $0.896$. However, we were given four possible answers, and we needed to determine which of these answers is closest to the value of $x$. We calculated the absolute differences between each answer and the value of $x$ and found that the answer $0.955$ is the closest.

The final answer is $\boxed{0.955}$.
Q&A: Logarithms and Equations

In our previous article, we explored the concept of logarithms and how to solve equations involving logarithms. We used the equation $\log _{\frac{3}{4}} 25 = 3x - 1$ to find the approximate value of $x$. In this article, we will answer some common questions related to logarithms and equations.

A logarithm is the inverse operation of exponentiation. In other words, if $a^b = c$, then $\log_a c = b$. The logarithm of a number is the exponent to which the base must be raised to produce that number.

To solve an equation involving a logarithm, you can start by isolating the logarithm term. You can do this by adding or subtracting the same value to both sides of the equation. Then, you can use the properties of logarithms to simplify the equation.

Some common properties of logarithms include:

• $\log_a b + \log_a c = \log_a (bc)$
• $\log_a b - \log_a c = \log_a \left(\frac{b}{c}\right)$
• $\log_a b^c = c \log_a b$

To find the value of a logarithm using a calculator, you can enter the equation into the calculator and press the "log" button. The calculator will give you the value of the logarithm.

A logarithmic function is the inverse of an exponential function. In other words, if $f(x) = a^x$, then $f^{-1}(x) = \log_a x$. The logarithmic function is used to solve equations involving exponential functions.

To graph a logarithmic function, you can use a graphing calculator or a graphing software. You can also use a table of values to create a graph.

Logarithms have many real-world applications, including:

• Finance: Logarithms are used to calculate interest rates and investment returns.
• Science: Logarithms are used to calculate the pH of a solution and the concentration of a substance.
• Engineering: Logarithms are used to calculate the power of a signal and the frequency of a wave.

To solve an equation involving a logarithm with a base other than 10, you can use the change of base formula:

$$\log_b a = \frac{\log_c a}{\log_c b}$$

where $c$ is any positive number other than 1.

In this article, we answered some common questions related to logarithms and equations. We covered topics such as the definition of a logarithm, how to solve an equation involving a logarithm, and the properties of logarithms. We also discussed real-world applications of logarithms and how to graph a logarithmic function.

The final answer is $\boxed{0.955}$.